Property Price Index Theory and Estimation : A Survey

Transcription

1 Grant-in-Aid for Scientific Research(S) Real Estate Markets, Financial Crisis, and Economic Growth : An Integrated Economic Approach Working Paper Series No.34 Property Price Index Theory and Estimation : A Survey Chihiro Shimizu and Koji Karato January, 2016 HIT-REFINED PROJECT Institute of Economic Research, Hitotsubashi University Naka 2-1, Kunitachi-city, Tokyo , JAPAN Tel: hit-refined-sec@ier.hit-u.ac.jp

2 Property Price Index Theory and Estimation: A Survey Chihiro Shimizu and Koji Karato February 12, 2016 Abstract Property has the particularity of being a non-homogeneous good, and based on this, it is necessary to perform quality adjustment when estimating property price indexes. Various methods of quality adjustment have been proposed and applied, such as the hedonic method often used in price statistics and, due to the fact that the information that can be used in estimation is limited, the repeat sales price method, methods using property appraisal price information, and so forth. However, since there is a lack of theoretical knowledge and data restrictions, it is no exaggeration to say that it is difficult to evaluate their practical application in the present situation. Therefore, focusing on the hedonic method that has been proposed as a quality adjustment method for property price indexes, in addition to repeat sales price indexes and indexes employing property appraisal prices, this paper aimed to outline the underlying econometric theory and clarify the advantages and disadvantages of the respective estimation methods. Key Words: Hedonic price index; Repeat sales price index; Age effect; Hybrid method; Property appraisal price method; SPAR Journal of Economic Literature Classification Numbers: C2, C23, C43, D12, E31, R21. This research was conducted by the authors while participating in the Hitotsubashi Project on Real Estate, Financial Crisis, and Economic Dynamics, (HIT-REFINED) supported by a JSPS Grant-in-Aid for Scientific Research (S). The authors thank Erwin Diewert, David Geltner and Marc Francke for helpful discussions. Professor, Institute of Real Estate Studies, National University of Singapore, cshimizu@nus.edu.sg. Professor, Faculty of Economics, University of Toyama. kkarato@eco.u-toyama.ac.jp. 1

3 1 Introduction The formation and collapse of property bubbles has a profound impact on the economic administration of many leading nations. The property bubble that began around the mid- 1980s in Japan has been called the 20th century s biggest bubble. In its aftermath, the country faced a period of long-term economic stagnation dubbed the lost decade. Many countries had similar experiences with this kind of problem for example, Sweden s economic crisis in the 1990s and the global financial crisis and economic stagnation caused by the formation and collapse of the U.S.-centered property bubble in the early 21st century. In light of this, it was pointed out that the information gap which existed between policymaking authorities and the property (including housing) and financial markets was a problem. In 2009, the IMF proposed the creation of a housing price index to the G20 in order to fill in this information gap, and the proposal was adopted. Furthermore, in 2011, it was suggested that the next economic crisis would be caused by land i.e., profit-generating property (commercial property) and it was decided to create a commercial property index as well. But how should these property price indexes be created? Property standards and facilities vary to a greater or lesser extent for each building, so there is no such thing as identical properties. Even if one assumed that the standards and facilities were the same, the process by which quality deteriorated would differ by building age, so the buildings would become non-homogeneous over time. In other words, property has the particularity of being a non-homogeneous good. In addition to this problem, the development of building technology is relatively fast, so quality changes over time. That is, not only does a building s functionality decline over time, but it becomes economically obsolescent with the advance of technology. As well, in cases where the surrounding environment changes significantly through redevelopment and the like, location characteristics such as transport accessibility of the city center also change. When attempting to capture temporal fluctuations in property prices while dealing with the problems caused by property being a non-homogeneous good and changes in quality, it is necessary to perform quality adjustment. In order to address these problems, there are quite a few points that can be adapted from existing index theory, as typified by consumer price statistics. For example, with regard to changes in quality accompanying technological development, the quality adjustment method known as the hedonic approach is used in private vehicle price statistics and the like. It would therefore be natural to also consider quality adjustment with the hedonic method for property price indexes, since this enables consistency with other types of economic statistics. However, when it comes to methods of quality adjustment for property price indexes, if one looks at the Residential Property Price Indices Handbook published by EuroStat in 2013, it present a variety of methods along with their advantages and disadvantages: a) Stratification or Mix Adjustment Methods b) Hedonic Regression Methods c) Repeat Sales Methods, and d) Appraisal-Based Methods. This is because, in reality, multiple methods have been applied in the estimation of property price indexes. Why have approaches other than the hedonic method been applied in practice? The first reason is the difficulty of quality adjustment. As explained previously, the reason for performing quality adjustment of property is that it is a good for which no homogeneity exists, so it is strongly heterogeneous. In such a case, in addition to the problems relating to quality changes faced in consumer price statistics and the like, one must also address said 2

4 heterogeneity. In other words, quality adjustment involves a high degree of difficulty. The second reason is the lack of usable price information at the micro level when estimating property price indexes. If attempting to apply the hedonic method, not only transaction price, transaction time, and land/building size but also location-related information such as the time to the city center and detailed information related to the building age and features are required. When there is no such information, the price index must be estimated with limited data. With the repeat sales method, quality adjustment is possible with just the transaction price and transaction time, so it has the advantage of minimizing the information needed with respect to property-related variables. That being the case, when attempting to measure price changes when information is limited, creating an index using only data for properties that are transacted repeatedly is consistent with the general thinking behind price index estimation methods. However, unlike other goods and service markets, the property transaction market is extremely thin. Thus, when attempting, for example, to create a monthly price index, one may easily anticipate that many problems will occur, since unlike markets where goods and services of identical quality are transacted frequently this is a market in which property with identical characteristics is transacted only once every few years. Third, in actual property transaction practice, it is not uncommon for property appraisal prices to be used. Not only is property strongly heterogeneous but there are few transactions depending on the region and usage, there are even markets where almost no transactions exist. In light of this, when trying to determine prices, using prices based on property appraisals is a valid approach. Thus, attempting to estimate property price indexes involves many difficulties. Compared to the housing market, where there is a relatively large number of transactions and the quality gap is small (i.e., it is more homogeneous), these difficulties mean that for commercial property (offices, retail facilities, hotels, logistics facilities, hospitals, farmland, etc.): 1) there will be more heterogeneity and a greater lack of information, 2) there will be a reduction in repeat sales samples, and 3) there is a greater probability of property appraisal prices being used. For markets for which property price information is relatively easy to obtain, the aim of this paper is to outline the characteristics of the hedonic method and repeat sales method that may be used in creating property price indexes, as well as estimation methods using property appraisal prices. Specifically, focusing on the hedonic method and repeat sales method, it will provide a comprehensive survey relating to quality adjustment methods when estimating property price indexes and clarify the characteristics of the various estimation methods. What s more, drawing on this outline, it will present a view of how property price indexes should be created, from the perspective of estimation method theory. 2 The Hedonic Price Method 2.1 The Hedonic Approach The hedonic approach is a technique established theoretically by Rosen (1974)[55]. Specifically, it treats a given product s price as an aggregate (bundle of attributes) of the values of the product s various attributes (characteristics) and estimates the various attribute prices using regression analysis. For many products circulating on the market, even when their intended use is the same, considerable differentiation exists based on their performance, functions, etc. Differences in attributes are reflected in the product s production costs. One could also say that consumer evaluations of the product s specific performance and functions are also 3

5 reflected in the price determined by the market. However, the attributes themselves are not necessarily bought and sold on the market. With the hedonic approach, by regressing product price on variables representing attribute quality and quantity, it is possible to measure the shadow price of non-market goods based on the estimated coefficient value. Lancaster (1966)[44] has conducted theoretical analysis of consumer behavior based on the assumption that consumer utility depends not on the product itself but on the various features, functions, etc., that comprise the product. The product s market price is thought to be determined based on supply and demand in relation to its various characteristics. However, the market with respect to these characteristics is not necessarily explicit but may be hidden in the background of product price determination. Lancaster s aim was to explicitly treat such underlying mechanisms and analyze consumer behavior in differentiated goods markets. Rigorously examining the relationship between differentiated product prices and consumer behavior is essential in preparing price indexes. For example, in the case of digital consumer electronics, passenger vehicles, housing, etc., even if the price is the same, quality will improve and functions increase as time passes. With the Laspeyres method, since a market basket is fixed at a baseline point in time, price indexes based on this method ignore changes in quality and functionality. Using the hedonic approach helps estimate the performance ratio between new and old products. Rosen (1974)[55] s price analysis of differentiated goods is a study that theoretically clarifies the manner in which product prices comprised by bundles of attributes are generated on the market. The study rigorously examines the relationship between the product supplier offer function, product demander bid function, and hedonic market price function, and characterizes the market price of products based on consumer and producer behavior. When this hedonic market price function is used, it is possible to obtain the acceptable payment amount for product attributes. Section 2.2 below outlines Rosen (1974)[55] s hedonic approach theory, while Section 2.3 addresses issues relating to the estimation of hedonic market price functions. Following Diewert, Heravi and Silver (2007)[27], Section 2.4 summarizes differences based on a hedonic dummy index and hedonic imputed index. Section 2.5 explains the characteristics of a producer price-related quality-adjusted hedonic index. Section 2.6 summarizes the characteristics of the hedonic price method. 2.2 Hedonic Approach Theory The Bid Function Following Rosen (1974)[55] s method, we will demonstrate the theoretical basis of the hedonic approach, using real estate as an example. The value of characteristic k comprising real estate shall be expressed as z k (k = 1, 2,..., K). Real estate characteristics represent size, building structure, kitchen, bathroom, accessibility of transportation, natural environment, social environment, and so forth. According to Rosen, the relationship between real estate market price p and characteristic value z 1,..., z k,..., z K may be expressed with the following hedonic price function h: p = h(z 1,..., z k,..., z K ) (1) The main objective of Rosen s analysis is to clarify how (1) is determined by the market. Given market price function (1), consumers select the optimal combination of real estate 4

6 characteristics. The issue of utility maximization may be formulated as follows: max U(x, z) x,z (2) s.t. I = x + h(z) (3) Here, U( ) is a well-behaved, strictly concave function, x is composite goods including goods and services other than real estate, z = (z 1,..., z k,..., z K ) is the real estate characteristic vector, and I is income. The composite goods price is standardized as 1. Based on the parameters of this optimization issue step, U k /U x = h k (z) is established. Note that U k = U(x,z) z k U x = U(x,z) x and h k (z) = h(z) z k. In other words, this shows that the marginal utility of the real estate characteristic measured using the marginal utility of income is equal to the marginal contribution value of the attribute in market prices. It is possible to determine the market price function using the bid function. Based on a given utility level u and income I, if the bid offered by a housing demander for real estate possessing characteristic z is taken as θ, then based on (2), this may be written as U(I θ, z) = u. If one solves this for θ, the amount that a consumer is able to spend on housing with respect to various combinations of characteristic z may be expressed as the bid function θ(z; I, u), given the utility level and income. In order to raise (lower) the utility level u, the bid for housing ( θ(z;i,u) u ) with characteristic z must decrease (increase) = Ux 1 < 0. Therefore, this shows that θ, when it reaches utility level u, is the maximum price that may be paid for housing with characteristic z. Based on (2), (3), and the bid function θ(z; I, u), one may write that U(I θ(z; I, u), z) = u. If this formula is partially differentiated for z k and 0 is included, the following is obtained: U x θ(z; I, u) z k + U k = 0 When the utility is maximized at the level of u, since U k /U x = h k (z ) for the optimal combination of characteristics z, the following two equations are definitely established: θ(z ; I, u ) = h k (z ) z k (4) θ(z ; I, u ) = h(z ) (5) (4) and (5) show that when the optimal characteristics are selected, the slope of the bid function and the slope of the market price function are consistent and the bid and market price are also equal. In other words, based on the optimal characteristic value, the bid function and market price function are contiguous. When consumer incomes and preferences vary, the bid function also varies. However, since the bid function and market price function must be contiguous in market equilibrium, the market price function is an envelope of the bid function for all consumers, with their various incomes and preferences The Offer Function It is also possible to define the price offer function for real estate suppliers and theorize the relationship with the market price function from the issue of profit maximization. For a given level of technology, the offer function is the minimum price offered when a given profit is reached. When a company selects the optimal characteristics and produces real estate, the 5

7 slope of the offer function and the slope of the market price (per unit of real estate) function will be consistent based on profit maximization behavior, and the offer price and market price will also be consistent. Therefore, based on the optimal characteristic value, the offer function and market price function are contiguous. Since heterogeneity exists in real estate producers technology, offer prices also vary in accordance with this. Since the offer price and market price need to be consistent in equilibrium, the market price function is an envelope of the offer function for various companies. Based on the above, the hedonic market price function is an envelope of both the bid function for an infinite number of real estate demanders and the offer function for an infinite number of real estate suppliers. As well, in the case of there being one supplier company, the bid function is equal to the marginal cost if one additional unit of real estate is produced (or the average cost per unit of real estate). As a result, the market price function is equal to the supplier s marginal cost Willingness to Pay If the bid function is used, it is possible to obtain consumers willingness to pay with respect to changes in attributes. For z, let us now assume that p = θ(z ; I, u ) = h(z ). When real estate K s characteristic zk is increased to (z K ), the demander s willingness to pay (W T P ) may be defined with the following formula: W T P θ ( z 1,..., z K 1, z K ; I, u ) p (6) In other words, when the characteristic value changes incrementally, the willingness to pay is the additional value that may be paid for real estate without changing the utility level. Since the utility function U is 2 z 2 k θ(z ; I, u ) = U 2 xu kk 2U x U k U xk + U 2 k U xx U 3 x < 0 when it is a strict concave function (the Hessian matrix is a negative definite matrix), the bid function is a concave function. (4) and (5) are established based on the optimal characteristic value combination z, and given that the bid function is a concave function, one can derive from θ ( z1,..., zk 1, zk ; I, u ) < h ( z1,..., zk 1, zk ) = p That p p > W T P (7) In other words, caution is required with regard to the limit value of market price function characteristics, since as long as demanders are not homogeneous, it is possible that the willingness to pay is overestimated. However, if it is assumed that changes in characteristic values will be sufficiently small, the market price function limit value may be used as an approximation of the willingness to pay. 2.3 Hedonic Market Price Function Estimation Function Types In order to accurately measure willingness to pay, estimation of the bid function is required, but in general an approximation is used by estimating the hedonic market price function (1). 6

8 When estimating the hedonic market price function, the function type is an issue. Given that simple estimation is possible, models such as double logarithms, semi-logs, and line shapes are often used. When real estate price at multiple points in time is observed as data, the hedonic market price at time t for property n may be described with the following formula: y t n = α t + z t nγ + ε t n (n = 1, 2,..., N(t); t = 0, 1,..., T ) (8) Here, y t n is the housing price logarithm (ln p t n) or exact numeric value (p t n), α t is the unknown time effect, z t n = (1, z t n1,..., z t nk,..., zt nk ) is the explanatory variable (characteristic) vector including a constant term, γ = (γ 0, γ 1,..., γ k,..., γ K ) is the coefficient vector, and ε t n is the error term. As an example, a semi-log model including the time effect may be written as: y t n = ln p t n = α t + γ 0 + K γ k znk t + ε t n (9) In this model, the estimation value of coefficient γ shows the effect of the characteristic value with respect to real estate price, and if a dummy variable is used for each point in time, estimation may be made estimated based on the method of least squares. To avoid multicollinearity and distinguish all parameters, it is necessary to perform some kind of standardization for α t and γ 0. Typically, at the observation starting point t = 0, it is considered that α 0 = 0, and a dummy variable for each point in time is used with respect to t = 1, 2,..., T. Since the function type of the hedonic market price function h cannot be specified in theoretical terms, it must be selected with a statistical technique. Even if specified in a double logarithmic model or semi-log model, the form is not necessarily the ideal one. Studies from the 1980s onward, such as Linneman (1980)[45], have performed non-linear estimation using Box-Cox transformation. In this case, the left side of (9) can be rewritten as follows: p λ 1 λ 0 y = λ (10) ln p λ = 0 Here, λ is an unknown parameter. Halvorson and Polakowski (1981)[36] tested various function forms by applying Box-Cox transformation to a flexible function form using a two-step approximation formula including a cross-term between explanatory variables. In response to their paper, Cassel and Mendelsohn (1985)[11] increased the explanatory power by including multiple cross-terms between variables, but pointed out that there is a reduction in the reliability of the coefficient estimation value due to multicollinearity and that interpretation of the marginal effect of hedonic characteristic values becomes more difficult. Cropper, Deck, and McConnell (1988)[13] performed statistical tests based on a translog form and Diewert-type utility function (Barten (1964)[3], Diewert (1971)[19], Diewert (1973)[20]), showing that if observational errors are included in the variables, a linear model or linear Box-Cox transformation model is superior to quadratic form Box-Cox transformation when it comes to formulation. There is also research that has proposed using a non-parametric method or semi-parametric method instead of a parametric function form to formulate the hedonic price function. With these approaches, attribute prices are inferred directly from the data without specifying a function form in advance (Knight et al. 1993[40], Anglin and Gencay (1996)[1], Pace 1995[53]). However, it has also been pointed out that, as with parametric analysis techniques, these do k=1 7

9 not free one from data-related problems (multicollinearity). In tests relating to the selection of parametric versus non-parametric models, Anglin and Gencay (1996)[1] have shown that it is relatively easy to dismiss parametric models. It is not that the parametric model variable structure is weak; rather, this result was demonstrated even for parametric models that passed a number of standard tests for model selection. Using a more flexible Generalized Additive Model (GAM), Pace (1998)[52] estimated a semi-parametric-type hedonic price function and demonstrated that it was superior to all types of parametric model. Since GAM itself is an established statistical technique, this finding shows that the incorporation of a non-parametric method in the hedonic approach is extremely effective The Problem of Distinguishing the Marginal Bid Value Function If characteristic values have a significant effect on market prices, the willingness to pay will cause divergence between the hedonic market price function and bid value function, so it is necessary to estimate the bid value function or bid value marginal effect. As a method of estimating the bid value function, Rosen (1974)[55] has proposed a method that regresses the market price function marginal effect on characteristic values and other exogenous variables. ĥ k = D k (z, A) (11) ĥ k = S k (z, B) (12) Here, ĥk is the marginal effect for hedonic market price function characteristic k, D( ) and S( ) are the characteristic s demand and supply functions, and A and B are vectors showing the real estate demander and supplier type, respectively (based on income, manufacturing technology, etc.). Since the marginal effect is the shadow price of the characteristic value, (11) and (12) are supply and demand simultaneous equations using inverse demand (bid value) and inverse supply (offer price), and supply and demand are distinguished using A and B as instrumental variables. Following Rosen s model, Witte, Sumka, and Erekson (1979)[?] estimated simultaneous equations for three characteristics covering multiple housing markets. However, as Brown and Rosen (1982)[6] have pointed out, it is not possible to properly distinguish between characteristic value supply and demand with estimation based on this method. Since the market price function marginal effect ĥk estimated in the first step is derived from h(z), one may consider that the characteristic price shown with the marginal effect is also a function of z. Demand for z depends on the various characteristic prices, and there is a correlation between characteristic prices and characteristic demand equation errors. In other words, it is possible that the effect of characteristic prices on characteristic demand is estimated with a bias. This problem of distinguishing the bid value function and offer function has been considered by Diamond and Smith (1985)[18] and Mendelsohn (1985)[49]. First, with regard to estimation of the first step hedonic market price function, it is pointed out that, apart from characteristic vectors, there is a need for exogenous variables not included in either the bid value function or offer function as well as for a characteristic value exponential term. Then, in the second step, a marginal bid value function simultaneous equation system is estimated simultaneously using exogenous variables solely to meet the distinction conditions. Sheppard (1999)[56] has discussed the distinction problem in greater detail. Ekeland, Heckman, and Nesheim (2004)[29] and Heckman, Matzkin, and Nesheim (2010)[39] proposed a distinction method for hedonic price estimation using a non-parametric approach. 8

10 2.4 Price Index Estimation Based on the Hedonic Approach Time Dummy Hedonic Regression The hedonic approach is a useful technique when creating quality-adjusted price indexes. There are two representative types of hedonic price index: (i) time dummy hedonic indexes and (ii) imputed hedonic indexes. Following Diewert, Heravi and Silver (2007)[27], we discuss differences between the two types of price index below. In (8), taking the observation period as two points in time, (t = 0, 1), one may assume the following estimation model that regresses logarithmic price on an explanatory variable vector with the time dummy and constant term excluded: y t n ln p t n = α t + K γ k znk t + ε t n (n = 1, 2,..., N(t); t = 0, 1) (13) k=1 Here, α t shows the average level of the product s quality-constant price for each period, and the overall scale of logarithmic price changes from time 0 to time 1 is α 1 α 0. Let us take 1 t as an N(t) dimension vector comprising everything from 1 and 0 t as an N(t) dimension vector comprising everything from 0. As well, let us take y 0 and y 1 as the N(0) and N(1) dimension vectors for the time 0 and time 1 logarithmic prices respectively, Z 0 and Z 1 as the N(t) K explanatory variable matrices for time 0 and time 1 respectively, and ε 0 and ε 1 as the N(0), N(1) dimension error vectors for time 0 and time 1 respectively. If we represent (13) as matrices for time 0 and time 1, they may be written as follows: y 0 = 1 0 α α 1 + Z 0 γ + ε 0 (14) y 1 = 0 1 α α 1 + Z 1 γ + ε 1 (15) Here, if we take α t, γ as estimators based on the method of least squares, one can formulate the following using the estimators and the realized value e t of the least squares residual error: y 0 = 1 0 α α 1 + Z 0 γ + e 0 (16) y 1 = 0 1 α α 1 + Z 1 γ + e 1 (17) For (16) and (17), if we define y = [ y 0 y 1 ] [ (N(0) + N(1)) 1 vector) e = e 0 e 1 ] (N(0)+N(1)) 1 vector), ϕ = [ [ ] α0 α1 γ ] 10 0 (2+K) 1 vector), and X = 0 Z Z 1 (N(0) + N(1)) (2 + K) matrix), then (16) and (17) may be rewritten as follows: y = Xϕ + e (18) Here, since X and the residual error e are orthogonal, we can obtain: X e = X (y Xϕ ) = 0 2+K (19) In other words, we can obtain 1 0e 0 = 0, 1 1e 1 = 0 and Z 0 e 0 + Z 1 e 1 = 0 K. Therefore, using the residual errors for (16) and (17): 1 0y 0 = N(0)α Z 0 γ (20) 1 1y 1 = N(1)α Z 1 γ (21) 9

11 If we work out α 0 and α 1 from this, we can obtain: α0 = 1 0y 0 N(0) 1 0Z 0 γ N(0) α1 = 1 1y 1 N(1) 1 1Z 1 γ N(1) = 1 0 = 1 1 ( y 0 Z 0 γ ) N(0) ( y 1 Z 1 γ ) N(1) (22) (23) (22) and (23) show the quality-constant logarithmic price level. 1 ty t /N(t) shows the arithmetic mean of the logarithmic price for time t = 0, 1 and 1 tz t /N(t) shows the arithmetic mean of the characteristic vector for time t = 0, 1. In other words, α 0 is equal to the result obtained by subtracting the average value of all characteristic values from the average value of the logarithmic price (arithmetic average of the quality-adjusted logarithmic price). Based on the above, the hedonic time dummy estimation value based on the logarithmic price change from time 0 to time 1 is the following differential: LP HD = α 1 α 0 (24) The explanatory variable matrices for (18) are expressed as follows: [ ] [ ] 10 0 V = 0 Z 0, Z = Z 1 Here, V is a (N(0)+N(1)) 2 matrix and Z is a (N(0)+N(1)) K matrix. If the explanatory variable is rewritten as X = [ V Z ], since the residual error vector is e = y Vα Zγ, the least squares estimator [ ] α α = 0 = (V V) 1 V (y Zγ ) (25) α 1 can be obtained from e e/ α = 0. Based on (25), the residual error may be rewritten as e = M (y Zγ ). Here, [ ] [ ] M 0 M = M 1 = I V(V V) 1 V I0 1 = 0 1 0/N(0) 0 0 I , 1/N(1) I is a (N(0)+N(1)) (N(0)+N(1)) identity matrix, and I t is a N(t) N(t) identity matrix. If we define y = My and Z = MZ, the error sum of squares is e e = (y Z γ ) (y Z γ ), so the least squares estimator for γ can be obtained as follows: γ = (Z Z ) 1 Z y = ( Z 0 Z 0 + Z 1 Z 1 ) 1 ( Z 0 y 0 + Z 1 y 1 ) (26) If we first calculate γ from (26) and then plug it into (25) (or (22) or (23)), the time effect estimator α can be obtained Imputed Hedonic Indexes Instead of performing estimation one time for two periods by pooling data, it is also possible to estimate the characteristic price parameter for each period. Taking η t as the N(t) 1 error term vector, the regression model for time t = 0 and time t = 1 may be written as follows: y t = 1 t β t + Z t γ t + η t (27) 10

12 Here, it is assumed that the characteristic price parameters γ 0, γ 1 vary depending on the observation period. If one includes β t and γ t as least squares estimators, the following formula may be established using the least squares residual error vector u t : y t = 1 t β t + Z t γ t + u t (28) Based on the nature of the residual error, [ 1 t Z t] u t = [ 0 0 K], so the following can be obtained: 1 0y 0 = N(0)β Z 0 γ 0 (29) 1 1y 1 = N(1)β Z 1 γ 1 (30) Therefore, the estimator for (29) and the time effect based on (29) can be obtained by solving these for β0, β1. ( β0 = 1 0y 0 N(0) 1 0Z 0 γ 0 = 1 0 y 0 Z 0 γ 0 ) (31) N(0) N(0) ( β1 = 1 1y 1 N(1) 1 1Z 1 γ 1 = 1 1 y 1 Z 1 γ 1 ) (32) N(1) N(1) The estimation value of the hedonic time dummy based on the logarithmic price change from time 0 to time 1 may be obtained using the differential LP HD = α1 α0. However, since it is assumed that the parameters γ 0, γ 1 for quality adjustment based on the formulation of (27) vary across periods for the two times, it is not possible to simply define the logarithmic price change from the differential of β0, β1. Therefore, Laspeyres type and Paasche type measures of price change in imputed hedonic model are shown as follows : [Laspeyres] φ L = (β Z 1 γ 1 ) (β Z 1 γ 0 ) (33) N(1) N(1) [Paasche] φ P = (β Z 0 γ 1 ) (β Z 0 γ 0 ) (34) N(0) N(0) where characteristic value change of a Laspeyres type measure is calculated as the arithmetical mean of Z 1, and that of a Paasche type measure is calculated as an arithmetical mean of Z 0. For both the differential φ L and φ P, adjustment with characteristic price is asymmetrical. Therefore, using the median value of the two differentials, the hedonic imputed estimation value based on the logarithmic price change from time 0 to time 1 is written as follows: LP HI = 1 2 φ L φ P = 1 1 { y 1 Z ( γ γ1 )} 1 0 N(1) { y 0 Z 0 ( 1 2 γ γ1 )} N(0) (35) Here, one can see that quality adjustment of price is performed not with Z t γ t but with Z ( t 1 2 γ γ1 ). If the sample sizes for the two times are identical and the characteristics and characteristic prices are constant over time, there is no difference between the two techniques. 11

13 2.4.3 Differences Between Time Dummy Indexes and Imputed Indexes In order to look at the differences between LP HD (24) and LP HI (35), the differential of the two may be expressed as follows: ( 1 LP HD LP HI = 1 Z 1 N(1) 1 0Z 0 ) ( 1 N(0) 2 γ0 + 1 ) 2 γ1 γ (36) In other words, if the average of the characteristic prices is equivalent for each time and if the pooled hedonic regression model characteristic price is equivalent to the hedonic characteristic price median value estimated for each time, (24) and (35) are fully equivalent. Based on (31) and (32), the β 0, β 1 least squares estimator regressed on each observation period is: β t = 1 t ( y Z t γ t ) /N(t) Using this, the least squares residual error may be written as follows: u t = M t y t M t Z t γ t (37) Here, M t = I 1 t 1 t/n(t). If we define y t = M t y t and Z t = M t Z t, the estimated characteristic price vector is as follows: γ t = ( Z t Z t ) 1 Z t y t (38) Here, if we multiply ( Z 0 Z 0 + Z 1 Z 1 ) by both sides of (26), the characteristic price using pooled data in (26) becomes: ( Z 0 Z 0 + Z 1 Z 1 ) γ = ( Z 0 y 0 + Z 1 y 1 ) = Z 0 Z 0 γ 0 + Z 1 Z 1 γ 1 (39) Note that, based on (38), (Z 0 Z 0 )γ 0 = Z 0 y 0 and (Z 1 Z 1 )γ 1 = Z 1 y 1. If γ 0 and γ 1 are equivalent, (39) shows that γ is necessarily the shared characteristic vector of these. If we multiply the right side of (36) ( 1 2 γ γ1 γ ) by 2 ( Z 0 Z 0 + Z 1 Z 1 ) from the right side, we obtain the following: 2 ( Z 0 Z 0 + Z 1 Z 1 ) ( 1 2 γ γ1 γ = Z 0 Z 0 γ 0 + Z 0 Z 0 γ 1 + Z 1 Z 1 γ 0 + Z 1 Z 1 γ 1 2 ( Z 0 Z 0 + Z 1 Z 1 ) γ ) = Z 0 Z 0 γ 0 + Z 0 Z 0 γ 1 + Z 1 Z 1 γ 0 + Z 1 Z 1 γ 1 2Z 0 Z 0 γ 0 2Z 1 Z 1 γ 1 = Z 0 Z 0 γ 0 + Z 0 Z 0 γ 1 + Z 1 Z 1 γ 0 Z 1 Z 1 γ 1 = ( Z 1 Z 1 Z 0 Z 0 ) ( γ 1 γ 0 ) In other words, ( 1 2 γ0 + 1 ) 2 γ1 γ = 1 ( Z 0 Z 0 + Z 1 Z 1 ) 1 ( Z 1 Z 1 Z 0 Z 0 ) ( γ 1 γ 0 ) (40) 2 If we plug (40) into (36), the differential of the time dummy index and imputed index shown with the logarithmic price difference may be rewritten using the following formula: LP HD LP HI = 1 ( 1 1 Z 1 2 N(1) 1 0Z 0 ) (Z 0 Z 0 + Z 1 Z 1 ) 1 ( Z 1 Z 1 Z 0 Z 0 ) ( γ 1 γ 0 ) (41) N(0) 12

14 Based on the above, when any of the following conditions are met, the two logarithmic price differentials based on the hedonic time dummy and hedonic imputation method are identical: 1 1 Z1 N(1) = 1 0 Z0 N(0) The average value of each characteristic is equivalent for the two times: The characteristic value variance/covariance matrices are equivalent for the two times: Z 1 Z 1 = Z 0 Z 0 The quality-adjusted prices obtained with the hedonic price estimation method for each time are identical: γ 1 = γ Summary of Hedonic Dummy Indexes and Hedonic Imputed Indexes As shown above, we have identified factors that show the differences between a hedonic dummy index and hedonic imputed index. In the regression equations, if it is possible to use information for two times and formulate the indexes with identical function forms, taking the (geometric) average of the two is perhaps a viable method when the two approaches show different results. However, rather than doing this, using either one index or the other is preferable for various reasons. A major issue of concern when using the hedonic time dummy (HD) method is that it has the following restriction: the characteristic price is fixed over time. However, the null hypothesis that the characteristic variable parameter is fixed throughout the observation period has in fact been dismissed by a number of papers. In contrast to this, the hedonic imputed index (HI) method is inherently more flexible than the time dummy model, which is a significant advantage. In Section 2.4.3, we showed that the difference between the two approaches depends on the following three variable factors: The characteristic average value The variance/covariance matrix of the characteristic value The estimated hedonic characteristic price What s more, multiplication of the difference between the two periods produces the ultimate difference. Therefore, the stability of the characteristic price parameter alone is not necessarily a problem. For example, even if the parameter is unstable, its instability will be alleviated by slight changes in other characteristics, and the same may be true for the price index. Due to the nature of the HD method, it uses independent variables observed for the two times, and it is restricted such that the characteristic price parameters are the same for the two times, and regression analysis ends up being executed one time only. In this sense, it may be said that the HD method is not flexible due to the presence of these restrictions. Why, then, are these restrictions imposed? Presumably, the reasons include the following: To not lose a degree of freedom. To provide an unambiguous estimation value for the overall price change from time 0 to time 1. To minimize the effect of abnormal values in conditions where there is a small degree of freedom. In contrast to this, the HI method allows for diachronic changes in characteristic prices and formulation is more flexible. However: A degree of freedom is lost. The estimation value for the overall price change in the two times is difficult to repro- 13

15 duce. Due to these and other issues, analysis costs increase. The latter of the two issues pointed out above may in fact not be all that serious, because Laspeyres- and Paasche-type estimation values for price changes are well established in index theory. Bearing these points in mind, the HI method may be considered the preferred method as long as the degree of freedom is not extremely restricted. In light of the above, the rolling window hedonic method that merges the hedonic dummy method and hedonic imputed method has been proposed. Market structural changes occur as a result of various exogenous shocks, but it is thought that a certain adjustment period exists until shocks are absorbed by the market and changes are realized. Therefore, the regression coefficient likewise does not change instantaneously but should instead be viewed as changing sequentially. However, if estimating a model where the data is divided into various periods and observation data for each period is used (as with the HI method), the links to prior and subsequent data are severed. As a result, under the assumption that structural changes occur sequentially, this method ends up making it more difficult to capture price changes within the sequential change process. Instead, as a more natural approach, a method of estimating price indexes within the sequential change process by taking an estimation period of a certain duration and estimating the model while moving this period as if obtaining a moving average may be preferable. A method that has been proposed based on this idea is the rolling window hedonic method. This approach is employed in the estimation of housing price indexes in Ireland and Japan. 2.5 Hedonic Production Price Index Measurement and Quality Adjustment The Producer Revenue Maximization Problem In this section, we will explain the characteristics of quality-adjusted hedonic indexes for producer prices, based on Diewert (2002)[24]. The Konus-type price index proposed in that paper is defined using a revenue function that is a value function of the revenue maximization problem based on company technology and resource constraints. The revenue function is derived from characteristic values constituting the product price, production technology, production factors, and product. We shall define the hedonic price (producer s willingness to pay) based on the characteristic vector z as: Π t (z) = ρ t f t (z) (42) Here, ρ t is the price showing the value of all characteristic values used for the product at time t and f t (z) shows the cardinal utility separable from the utility function. In (42), it is assumed that the utility function is equivalent for the two times. f 0 = f 1 (43) Given the hedonic price (42), the producer performs revenue maximization. First, we shall define the production function F as follows: q = F t (z, v) (44) Here, q is the production volume and v is the production factor vector. For a given level of 14

16 production technology, the following revenue-maximizing value function may be obtained: R(ρ s f s, F t, Z t {, v) max ρ s f s (z)q : q = F t (z, v); z Z t} q,z { = max ρ s f s (z)f t (z, v); z Z t} (45) z Here, Z t shows the feasible set of characteristic values. When the characteristics and input factors for time t are taken as z t, v t, the corresponding production volume is: q t = F t (z t, v t ) (46) Therefore, the maximized revenue function for time t may be written as follows: R(ρ t f t, F t, Z t, v t { ) max ρ t f t (z)q : q = F t (z, v t ); z Z t} q,z = ρ t f t (z t )q t ; t = 0, 1 (47) Konus-Type Hedonic Production Price Indexes Using the maximized revenue function (47), the Konus-type hedonic product price index between time 0 and time 1 is defined as follows: P (ρ 0 f 0, ρ 1 f 1, F t, Z t, v) = R(ρ1 f 1, F t, Z t, v) R(ρ 0 f 0, F t, Z t, v) The differences between the two revenue functions are caused by the hedonic prices ρ 1 f 1 and ρ 0 f 0. Since max z { ρ 1 f 1 (z)f t (z, v t ); z Z t} = max z { ρ 1 f 0 (z)f t (z, v t ); z Z t} based on hypothesis (43), (48) may be rewritten as follows: (48) P (ρ 0 f 0, ρ 1 f 1, F t, Z t, v) = ρ1 R(ρ 0 f 0, F t, Z t, v) ρ 0 R(ρ 0 f 0, F t, Z t, v) = ρ1 ρ 0 (49) In estimation of the hedonic price, if we assume that the utility of the characteristic portion is diachronically constant, the Konus-type product price index may be estimated very easily. Let us consider general cases that do not meet hypothesis (43). Taking the price index in (49) as our base, we can define an observable hedonic Laspeyres production price index and Paasche production price index with the following inequalities, using: P (ρ 0 f 0, ρ 1 f 1, F 0, Z 0, v 0 ) = R(ρ1 f 1, F 0, Z 0, v 0 ) R(ρ 0 f 0, F 0, Z 0, v 0 ) ρ1 f 1 (z 0 ) ρ 0 f 0 (z 0 ) = P HL (50) P (ρ 0 f 0, ρ 1 f 1, F 1, Z 1, v 1 ) = R(ρ1 f 1, F 1, Z 1, v 1 ) R(ρ 0 f 0, F 1, Z 1, v 1 ) ρ1 f 1 (z 1 ) ρ 0 f 0 (z 1 ) = P HP (51) Here, P (ρ 0 f 0, ρ 1 f 1, F 0, Z 0, v 0 ) and P (ρ 0 f 0, ρ 1 f 1, F 1, Z 1, v 1 ) are theoretical production price indexes that cannot be observed. (50) shows that the theoretical production price index P (ρ 0 f 0, ρ 1 f 1, F 0, Z 0, v 0 ) has the observable Laspeyres production price index P HL as its lower limit, and (51) shows that the theoretical production price index P (ρ 0 f 0, ρ 1 f 1, F 1, Z 1, v 1 ) has the observable Paasche production price index P HP as its upper limit. By using these convex combination equations (weighted averages) instead of F 0, Z 0, v 0 or F 1, Z 1, v 1 constituting the production price index, it is possible to define the range that can 15

17 be covered by the theoretical Laspeyres production price index and Paasche production price index. If the scalar λ [0, 1] is used, the convex combinations for F t, Z t, v t for period t = 0, 1 may be written as follows. Z(λ) = (1 λ)z 0 + λz 1 v(λ) = (1 λ)v 0 + λv 1 F (λ) = (1 λ)f 0 (z, v(λ)) + λf 1 (z, v(λ)) Therefore, the hedonic production price function may be written as: P (λ) = R(ρ1 f 1, F (λ), Z(λ), v(λ)) R(ρ 0 f 0, F (λ), Z(λ), v(λ)) = max { z ρ 1 f 1 (z)f (λ); z Z(λ) } max z {ρ 0 f 0 (z)f (λ); z Z(λ)} (52) When λ = 0 since P (λ) signifies that P (ρ 0 f 0, ρ 1 f 1, F 0, Z 0, v 0 ) the following may be derived from inequality (50): P (0) P HL = ρ1 f 1 (z 0 ) ρ 0 f 0 (z 0 (53) ) As well, when λ = 1, since P (λ) signifies that P (ρ 0 f 0, ρ 1 f 1, F 1, Z 1, v 1 ), the following may be derived from inequality (51): P (1) P HP = ρ1 f 1 (z 1 ) ρ 0 f 0 (z 1 (54) ) By using Diewert s proof (1983; )[22], if P (λ) is a continuous function for [0, 1], it is possible to show that λ exists, whereby 0 λ 1 and P HL P (λ ) P HP P HP P (λ ) P HL. In other words, one can see that the theoretical hedonic production price index for the period t = 0, 1, when considered via P (λ ) described above, exists between the observable Laspeyres production price index and Paasche production price index. Note that to obtain this result, one must assume the continuity of λ in the hedonic model price functions ρ 1 f 1 (z 0 ), ρ 0 f 0 (z 0 ) in the numerator and denominator of Formula (52), the production functions F 0 (z, v), F 1 (z, v), and the feasible characteristic value sets Z 0, Z 1. The sufficient conditions for continuity are: The production functions F 0 (z, v), F 1 (z, v) are positive and continuous for z and v. The hedonic model price functions f 0 (z), f 1 (z) are positive and continuous for z. ρ 0, ρ 1 are positive. Sets Z 0, Z 1 are convex sets, bounded, and closed. Based on the above, one can see that the boundary range for the theoretical price index is determined by the observable price index. In order to obtain the best value for approximating the theoretical index, it is natural to take the adjusted average of the two boundary values. If the adjusted average function for the Laspeyres and Paasche production price indexes is written as m(p HL, P HP ), we can confirm, based on Diewert s argument (1997; 138)[23], that m() must be the geometric average. In other words, the best candidate in terms of approximating the theoretical production price index is the following observable Fisher hedonic production price index, using (50) and (51): P HF = (P HL P HP ) 1/2 = ρ1 ρ 0 16 ( f 1 (z 0 ) f 0 (z 0 ) ) 1/2 ( f 1 (z 1 ) 1/2 ) f 0 (z 1 )

18 If the hypothesis f 0 = f 1 is fulfilled by the hedonic model price function being the same for the two times, then this can be transformed into P HF = ρ 1 /ρ 0. As well, if the respective observable prices are defined as P 0 = ρ 0 f 0 (z 0 ) and P 1 = ρ 1 f 1 (z 1 ) (55) the Laspeyres and Paasche production price indexes can be shown as quality-adjusted price comparisons: P HL = ρ1 f 1 (z 0 ) ρ 0 f 0 (z 0 ) = P 1 /f 1 (z 1 ) P 0 /f 1 (z 0 ) P HP = ρ1 f 1 (z 1 ) ρ 0 f 0 (z 1 ) = P 1 /f 0 (z 1 ) P 0 /f 0 (z 0 ) Therefore, the Fisher hedonic production price index may be written as follows: (56) (57) P HF = (P HL P HP ) 1/2 = ( P 1 /f 1 (z 1 ) 1/2 ( ) P 1 /f 0 (z 1 ) 1/2 ) P 0 /f 1 (z 0 ) P 0 /f 0 (z 0 (58) ) In other words, the Fisher hedonic production price index may be obtained from the geometric average of the two quality-adjusted price indexes obtained by estimating the hedonic regression model. The hedonic approach is useful not just for quality adjustment of the product user price but also for quality adjustment of the product supplier price. In this chapter, in order to define a product price index assuming competitive company production activities, we used a revenue function (total willingness to pay) maximized based on Konus. If the cardinal utility function for the characteristic portion is the same at the two points in time, the theoretical production price index may be shown by comparison with the observable product price. In addition, in general cases, based on certain restrictions, we showed that the theoretical production price index is present in the range that forms the boundary values of the observable Laspeyres and Paasche production price indexes. 2.6 Characteristics, Advantages, and Disadvantages of the Hedonic Method Rosen (1974)[55] developed a market equilibrium theory for differentiated products. This study rigorously examined the relationship between the structures of the product supplier offer function, product demander bid value function, and hedonic market price function, and characterized product market price based on consumer and producer behavior. If the bid value function is used, it is possible to obtain the consumer s willingness to pay with respect to changes in characteristics. In market equilibrium, not only are the market price and bid value consistent, but the slope of the hedonic function and bid value function are also consistent. Since the bid value function is a concave function, the willingness to pay with respect to incremental changes in characteristics is smaller than the change in the market price. In other words, caution is required with respect to the market price function characteristic limit value, since it is possible that the willingness to pay will be overestimated as long as demanders are not homogeneous. However, if one assumes that changes in characteristic values will be sufficiently small, the market price function limit value may be used as an approximation of the willingness to pay. Therefore, the market price function is generally estimated in existing research. If the hedonic approach is used, it is possible to measure changes in quality-adjusted price using samples at another point in time. The simplest and most widely used method is to estimate 17